Mini Ghosh

Modeling the spread of HIV in a stage structured population: Effect of awareness.

Human immunodeficiency virus (HIV) is a lenti-virus (a member of the retrovirus family) that causes acquired immunodeficiency syndrome (AIDS), a critical condition in humans in which progressive failure of the immune system allows life-threatening opportunistic infections. Over the past few years HIV has been spreading rapidly in the population. Almost, everyday there are thousands of new human cases of HIV infection being recorded in the world and these occur in almost every country of the world. However, the spread of HIV is relatively faster in the developing countries as compared to developed countries because developing countries have limited resources. Worldwide, 70% of HIV infections in the adults have been transmitted through heterosexual contact and vertical transmission accounts for more than 90% of global infection in infants and children. In this paper, we propose a nonlinear mathematical model to study the spread of HIV by considering transmission of disease by heterosexual contact and vertical transmission. A stage structured model is proposed and analyzed by considering the total population variable and dividing the whole population under consideration into three stages: children, adults and old. Also, in this paper it is assumed that the rates of recruitment are different in different groups of population. Equilibria of the model and their stability are also discussed. Using the stability theory of differential equations and computer simulation, it is shown that due to the increase in the awareness of the disease in the adult class the total infective population decreases in the region under consideration.

MODELING PREY-PREDATOR TYPE FISHERY WITH RESERVE AREA

This paper presents a prey-predator type fishery model in two patch environment. We assume that one patch is accessible to both prey and predator while other is a refuge for prey. The prey refuge constitutes a reserve zone where fishing is not allowed. In the unreserved area there are both prey and predators and harvesting of only predators is permitted. The equilibria of the model and their stability are discussed. Also we investigate various possibilities of bionomic equilibrium. Finally, the model is simulated for varied set of parameters and sensitivity of the parameters and effect of the size of reserve on maximum yield are studied. We found that it is the best to keep the size of reserve small and increase the harvesting effort upto optimal harvesting effort to get the higher maximum yield.

COEXISTENCE OF THE STRAINS INDUCED BY MUTATION

In this paper, we propose a two strain epidemic model with single host population. It is assumed that strain one can mutate into strain two. Also latent-stage progression age and mutation are incorporated into the model. Stability of equilibria (including the disease free equilibrium, dominant equilibria and the coexistence equilibrium) is investigated and it is found that they are locally stable under suitable and biological feasible constraints. Results indicate that the competition exclusion and coexistence of the two strains are possible depending on the mutation. Numerical simulations are also performed to illustrate these results.

Spread of Zika virus disease on complex network—A mathematical study

Abstract Zika virus epidemic poses a major threat to public health globally. Initially, this disease was limited to Africa but now it is spreading throughout the World. It is well known that zika virus is transmitted to human by the bites of Aedes mosquitoes. Recently, there are reported cases of sexual transmission and transmission due to blood transfusion in many countries e.g., Argentina, France, New Zealand, USA etc. In this paper a mathematical model of Zika virus on complex network is formulated and analyzed keeping in mind both mosquito-borne and sexual transmission of this disease. The existence and global stability of disease-free equilibrium are discussed in detail. The basic reproduction number R 0 of the model is computed and it is found that for R 0 1 , the disease-free equilibrium of the model is globally asymptotically stable under some condition. In addition to this the final size relation of the proposed model is also computed. The key parameters of the model are computed using curve fitting to the real data by least-square method. Sensitivity analysis and numerical simulation are also performed for our proposed model. Finally, this model is extended to optimal control problem, to find the suitable cost-effective and time-dependent control strategies to reduce the number of infectives in a desired interval of time.

Mathematical modeling of zika virus disease with nonlinear incidence and optimal control

The Zika virus was first discovered in a rhesus monkey in the Zika Forest of Uganda in 1947, and it was isolated from humans in Nigeria in 1952. Zika virus disease is primarily a mosquito-borne disease, which is transmitted to human primarily through the bite of an infected Aedes species mosquito. However, there is documented evidence of sexual transmission of this disease too. In this paper, a nonlinear mathematical model for Zika virus by considering nonlinear incidence is formulated and analyzed. The equilibria and the basic reproduction number (R0) of the model are found. The stability of the different equilibria of the model is discussed in detail. When the basic reproduction number R0 1, we have endemic equilibrium which is locally stable under some restriction on parameters. Further this model is extended to optimal control model and is analyzed by using Pontryagins Maximum Principle. It has been observed that optimal control plays a significant role in reducing the number of zika infectives. Finally, numerical simulation is performed to illustrate the analytical findings.

Modeling and analysis of the symptomatic and asymptomatic infections of swine flu with optimal control

Swine flu is an infectious disease which spreads very rapidly in the population. Infected droplets are expelled into the air by swine flu infected individuals through coughing and sneezing. This disease is transmitted to susceptible individuals by inhalation or ingestion of these infected droplets containing virus. In this paper, we propose and analyze a mathematical model for Swine Flu by considering symptomatic and asymptomatic infections. It is assumed that the transmission rates due to symptomatic and asymptomatic individuals are different. The mathematical model is formulated by assuming simple mass-action type incidence. The basic reproduction number R0 of the model is computed and the local and the global stabilities of different equilibria of the model are studied. Further, this model is extended to optimal control model. The optimal control model is analyzed using Pontryagin’s Maximum Principle and is solved numerically using MATLAB. Finally numerical simulation is performed to see the effect of optimal control on the infected population. It is observed that optimal control model gives better result compared to the model without optimal control as it reduces the number of infectives significantly in a desired interval of time.

The role of screening and treatment in the transmission dynamics of HIV/AIDS and tuberculosis co-infection: a mathematical study

In this paper, we present a deterministic non-linear mathematical model for the transmission dynamics of HIV and TB co-infection and analyze it in the presence of screening and treatment. The equilibria of the model are computed and stability of these equilibria is discussed. The basic reproduction numbers corresponding to both HIV and TB are found and we show that the disease-free equilibrium is stable only when the basic reproduction numbers for both the diseases are less than one. When both the reproduction numbers are greater than one, the co-infection equilibrium point may exist. The co-infection equilibrium is found to be locally stable whenever it exists. The TB-only and HIV-only equilibria are locally asymptotically stable under some restriction on parameters. We present numerical simulation results to support the analytical findings. We observe that screening with proper counseling of HIV infectives results in a significant reduction of the number of individuals progressing to HIV. Additionally, the screening of TB reduces the infection prevalence of TB disease. The results reported in this paper clearly indicate that proper screening and counseling can check the spread of HIV and TB diseases and effective control strategies can be formulated around ‘screening with proper counseling’.